Multi-Agent Plausibility Models

A multi-agent plausibility model is a structure M=hW,≤a, RC, Via∈A,C∈C, whereAis a finite set of agentsa1, ..., an,≤a is the plausibility order for each

agent a∈ Aas defined before and just indexed by the nameai for the agent.

We define everything, such as bestS etc., as before but index it with a for each agent, i.e. for S ⊆W, bestaS ={v ∈ W|∀x∈ S : x≤a v}. Truth at a

world in a model is defined as before since all operators are independent of the agents.

When we have a multi-agent plausibility model M, we define a doxastic accessibility relation →a. Let for w ∈ W, w(a) = {v ∈ W|v 'a w}, i.e.

the set of worlds v which a considers to be equally plausible to w. Then for s, t∈W:

s→a t⇔t∈max≤as(a)⇔t∈ {v ∈W|∀x∈s(a) :x≤a v}

That is t is doxastically accessible from s by agent a iff. t is at least as plausible as any world v considered equally plausible to s by agent a.

The overt beliefs Lewis uses in his analysis we treat as common beliefs, and hence will be a stricter concept than the one Lewis mentioned. Common belief ofϕcan be understood as ‘everybody believesϕand everybody believes that everybody believesϕand so on’ as opposed toalmosteverybody believes etc.

As Bonanno (1996) emphasizes, from a semantic perspective, common belief understood this way is unproblematic. However syntactically, it poses some problems due to being an infinite conjunction and most usually used formal languages are finitary. Since our approach is semantic, we will not address the syntactic issues here. According to Bonanno (1996), semantically,

the common belief accessibility relation →CB for a group of agents G ⊆ A

is the transitive closure of the union of the individual accessibility relations

→a, that is, it is the smallest transitive relation→, s.t. S_a∈G →a⊆→. Thus,

a formula ϕ is commonly believed at a world w by group G iff. ϕ is true at any world v, s.t. w→CBG v. We define this formally, but as an operator in

our metalanguage, for w∈P:

w+_{CBGϕ iff. for anyv ∈W :w→}

CBG v implies v

+ _ϕ

w−CBGϕ iff. for somev ∈W :w→CBG v implies v

+_ϕ

Forw∈I, we again assign the truth value arbitrarily. Note thatw+ _CB

Gϕ entails that for any a∈ G, ϕ∈ Ba.

Proof. Consider an arbitrary multi-agent plausibility model M = hW,≤a

, RC, Via∈A,C∈C. Suppose w + CbGϕ and let v ∈ bestaW be arbitrary for

arbitrarya∈ G. Thenw≤a v. Hence v ∈max≤aw(a) and thus w→av and

thus w →CB v. Therefore v + ϕ and thus ∀x ∈ bestaW : x + ϕ. Hence

ϕ∈ Ba.

This is reasonable since everything that is commonly believed by a group should also be believed by every single member of the group.

How are we to understand a common belief world then? What Lewis has in mind is not all the worlds where the common beliefs at w are also commonly believed, but where the common beliefs from w are true. Hence, a common belief world isnot just a world in_JCBw

GK, where CB

w

G ={ϕ|w+ CBGϕ}. Instead, for Lewis (1978, p. 44), a common belief world is certainly a world at which all common beliefs of a certain community are true:

Then we can assign to the community a set of possible worlds, called the collective belief worlds [i.e. common belief worlds] of the community, comprising exactly those worlds where the overt beliefs all come true.

In our case, this community is the community of origin of a fictionf. Usually, this community is a community which existed in the past of “our” world, or, more generally, part of the world at which we are to evaluate ‘Inf, ϕ. We use

‘our’ instead of ‘actual’, since Lewis understands ‘actual’ as an indexical. So on his account ouractual world, is not the same as theactual world of, say, the community of origin ofLOTR. However, our world, without the proviso ‘actual’, is spatio-temporally inclusive on Lewis’s (1986) account and hence comprises the community of origin of LOTR. Hence, their common belief worlds, will be worlds in which all the common beliefs they had in our world, are true.

If CBw

G ={ϕ|w+ CBGϕ} is the set of common beliefs of group G atw, then we let |CBw

G| be the set of worlds, where all those common beliefs are true. Formally, this amounts to

|CBGw|={v ∈W|∀ϕ∈CBGw :v + ϕ}= \ ϕ∈CBw G JϕK +

We now go on to discuss the revision on the multi-agent model.